This section is devoted to the proof of Theorem REF . We begin by showing an analogous statement for stream functions belonging to \(H^2\) , see Theorem REF  below. Afterwards, we will use elliptic regularity theory to improve the regularity of the constructed steady solutions up to real analytic, see Theorem REF  below and section REF . We follow the strategy presented in [1]} to prove the existence of non-trivial non-shear solutions to the Euler equation arbitrarily close to the Kolmogorov flow on the square torus.
